MDM4U Name _______________________________ MDM4U Exam Dec 20 and 21, 2017 Exam Review Package Exam Review Types of data Sources of bias Sample size Sampling techniques Population vs. sample Characteristics of a good sample Characteristics of an Effective Survey Ethics Privacy Need for honesty No Bias Design Questionnaires/Surveys Analysis of one-variable data Mean, median, mode Range, interquartile range Variance, standard deviation Quartiles, percentiles and z-scores Graphs for 1V data Analysis of 2V data scatterplots Correlation coefficient Coefficient of determination Lines of best fit/linear regression Types of relationships between 2V Statistics in the media Probabilities used to represent likelihood of an experiment Venn Diagrams Sample space Union of events Intersection of events Theoretical probability Experimental probability Conditional Probability, dependent events Mutually exclusive events Permutations Combinations Additive and multiplicative counting principle Pascal’s triangle and combinations Probability histogram Expected value Discrete Probability Distributions o Binomial o Hypergeometric o Geometric o Uniform Discrete vs. continuous frequency histograms Normal distributions Normal Approximation of Binomial Distributions z-scores Margin of error and confidence intervals 1. Identify each of the following variables as qualitative or quantitative. For each quantitative variable, identify whether it is continuous or discrete, and for qualitative variables, identify whether it is nominal or ordinal. (a) eye colour (b) number of candies in a bag (c) volume of a drink (d) time of a ball’s decent (e) colour of light (f) carpet texture (g) thickness of a book (h) day of the week 2. Identify the variables and their types, as well as the population for the following thesis questions. What type of sampling would you use in each case? Explain. (a) How many students in your Grade 12 math class have a part-time job? (b) Is there a relationship between homework completion and test performance in a Grade 10 science class? (c) Do more families take a vacation during winter break or spring break in your community? (d) Are the females or males in your high school more likely to smoke? (e) Are teens in Ontario more likely than adults to drink and drive? 3. Identify the type of sampling technique used in each of the following scenarios. (a) John picks various coloured marbles out of a box. (b) The Ontario government randomly selects five high schools in Ontario and surveys every teacher in those schools. (c) Every fiftieth family in the London telephone book is surveyed by phone. (d) Before booking a DJ for a school dance, the student council surveys 25% of the students from each grade. (e) Ramandeep conducts a survey regarding cyber bullying with students that pass his classroom. MDM4U Name _______________________________ MDM4U Exam Dec 20 and 21, 2017 Exam Review Package 4. The following is a list of species for five types of animals Mammals Reptiles Amphibians cheetah turtle frog elephant alligator salamander dog rattlesnake toad human lizard mud rabbit crocodile newt (a) Select 6 animals using simple random sampling. (b) Select 5 animals using systematic random sampling. (c) Select 6 animals using stratified random sampling. Fish shark stingray trout eel carp Birds woodpecker lark goose ostrich chicken 5. Identify the type(s) of bias that might result from each of the following data collection methods. How would you change the data collection method to avoid this bias (a) You email surveys to your classmates to be returned to you next week. (b) You are interested in the study habits of grade 12 students, so you interview students from your class. (c) You ask students about their recycling habits on behalf of the MTS Save the Planet Team. (d) You take a random sample of students in the cafeteria during second lunch to determine their attitudes towards a new school attendance policy. 6. Write a proper survey question to collect the following data. (a) the respondent’s opinion about the food served at the school cafeteria (b) the respondent’s ideas about a career choice (c) the respondent’s thoughts about alternative energy sources in Canada’s future (d) the respondent’s favourite sport to watch (e) the respondent’s confidence in his or her mathematical ability The data in the table below represent odometer readings of used cars on a used-car lot. Use the data to answer questions 7 to 12. 56 95 128 102 76 113 87 115 65 109 176 105 102 57 43 98 102 128 81 92 34 144 104 95 69 84 155 137 73 223 78 128 117 116 192 120 7. a) Group these data into intervals and create a frequency table. b) Produce a frequency histogram. 8. a) Determine the three measures of central tendency using the raw data.. b) Determine the three measures of central tendency using the midpoints of the intervals. c) Compare the results of parts a) and b). Explain the differences. 9. Use a graphing calculator to determine the population standard deviation and the interquartile range using the raw data. 10. a) Produce a box-and-whisker plot b) Identify any outliers. c) How have the outliers affected the measures of central tendency? 11. The used-car dealer uses the median to describe the average odometer reading. Is this appropriate? Explain. 12. The dealer intends to sell the cars in the 20th percentile at premium prices. To which odometer readings would this apply? MDM4U Name _______________________________ MDM4U Exam Dec 20 and 21, 2017 Exam Review Package 13. The final mark in a data management class is calculated using the weightings shown in the table below. What was Sarah’s final mark? Weighting Sarah’s Mark 10% Quizzes 78 out of 84 10% Assignments 102 out of 120 Tests 30% 203 out of 250 Culminating Assignment 20% 85 out of 100 Exam 30% 76 out of 90 14. Describe the type of sampling technique used in each scenario. a) a telephone survey of numbers randomly selected from a store’s database. b) A student council asks students to comment on an issue by placing their comments in a comment box. c) A proportionate number of people from 10 age groups are randomly selected from the population of a city. d) A pollster randomly selects three streets from 10 randomly selected towns in Ontario and interviews a resident of the street. 15. Identify the bias in each survey scenario. Suggest how to eliminate the bias. a) A teacher asks the boys in the grade 9 health class to raise their hands for how many dates each student has had as the teacher calls out the numbers. b) A magazine asks readers to respond to a poll about their favourite actor of the Academy Awards nominees, immediately following an article about one of the actors. c) A pollster asks pedestrians on a downtown sidewalk if they are in favour of a new park in the suburbs. 16. Define or explain each of the following terms. a) moderate positive linear correlation b) r2 c) accidental relationship d) correlation coefficient e) coefficient of determination 17. Match the correlation type with the coefficient, r. Correlation Type Coefficient, r a) weak, positive, linear 0.00 b) strong, negative, linear 1.00 c) none 0.46 d) perfect, positive, linear 0.20 e) moderate, negative, linear 0.75 18. In a physics experiment, various masses were suspended from a spring. The stretching distance of the spring is recorded for each case. 50 100 150 200 300 500 750 Mass (g) 1.2 2.5 3.2 5.3 7.0 11 16 Stretch Distance (cm) a) Create a scatter plot and classify the linear correlation. b) Determine the correlation coefficient and line of best fit (using technology) d) Use this model to predict the stretch length of this spring when loaded with a 625-g mass. Is this interpolation or extrapolation. MDM4U Exam Review Package Name _______________________________ MDM4U Exam Dec 20 and 21, 2017 19. Glyn has a spinner with two colours, red and black. He spins it four times in a row. a) Calculate the number of different orders in which the spinner could land red or black. b) Draw a tree diagram to illustrate all the possible results. c) Explain how your tree diagram corresponds to your calculations in part a)? 20. You and your friend are designing a method whereby you can communicate across a crowded room with hand signals. The three signals are hand open with palm facing the other person, hand open with back of hand facing the other person, and closed hand. a) How many different signals can you define using i) 3 signals? ii) 1, 2, or 3 signals? 21. a) How many four-digit numbers are possible with the digits 3, 4, 5, 6, 7, 8, and 9? Assume that no digits repeat. b) How many of these numbers in part a) are odd numbers? c) How many of these numbers in part a) are even numbers? 22. How many ways are there to roll either a 4 or a 9 with a pair of dice? 23. How many different ‘words’ can you form with the letters in each of the following words: a) MATH b) CAREER c) EXTINCT 24. How many 4-digit numbers less than 8000 have at least one 9, if repetition is allowed? 25. On the checkerboard shown, the piece can travel only diagonally upward. It can’t move through the squares containing an explosion, but can jump over them! Determine the number of paths from the checker’s current position to the top of the board. 26. Loyalist CVI is about to have a reunion of all the previous graduating classes. There are 20 members of the current alumni association. Of these, eight members are recent graduates. In how many ways can a sub-committee of four members be struck to provide memorabilia if a) there are no restrictions b) the sub-committee must be all recent graduates c) the sub-committee must have only two recent graduates d) the sub-committee must have no more than three recent graduates 27. There are 15 part-time workers at a local grocery store. a) In how many ways can the general manager choose three part-time workers to attend a seminar? b) In how many ways can the general manager choose a clerk, a stock person, and a bakery clerk? c) Should the answers to parts a) and b) be the same? Explain. 28. A soccer team played eight games and won five of them. There were no ties. How many arrangements of the five wins and three losses are possible? 29. There are 55 students who take Data Management this term and 24 who take Calculus. Only 5 students take both courses. A committee of 4 students is to be formed to represent this group. In how many ways can this committee be formed if there must be at least one student from Data Management? MDM4U Name _______________________________ MDM4U Exam Dec 20 and 21, 2017 Exam Review Package 30. A school athletics department fields a number of teams. 15 students play on the volleyball team 18 students play on the basketball team 21 student play on the soccer team 8 students play both volleyball and basketball 6 students play both volleyball and soccer 7 students play both basketball and soccer 4 students play on all three teams a) Use a Venn diagram to determine the minimum number of trophies that must be ordered if each student is to receive a commemorative trophy? b) How many students played volleyball but neither basketball nor soccer? 31. Every Friday a downtown restaurant has an all-you-can-eat buffet with the following items: soup, garden salad bread, rolls, crackers popcorn shrimp, shrimp and dip lasagna, chicken wraps, beef stew snow peas, carrots, mixed vegetables fresh fruit, apple pie, custard a) How many different combinations of items could you choose for your meal? b) The restaurant also has a lunch special featuring any one item from each group. How many choices do you have in this special? 32. The game of euchre uses just the 9s, 10s, jacks, queens, kings, and aces from a standard deck of 52 cards. How many five-card euchre hands have at least one spade? 33. A pencil case contains three blue pens, two red pens, and five pencils. If you reach in and randomly select a writing instrument, what is the probability that it is a) a red pen? b) a pen? c) not a blue pen? 34. Lena programs her graphing calculator to generate a random number between 1 and 10, and conducts 20 trials. Let E be the event that an even number is generated, and that she observes that this occurs seven times. a) What is the experimental probability that E occurs? b) What is the theoretical probability that E occurs? c) Lena explains that the random number generator on her graphing calculator is flawed. What is Lena’s reasoning? d) What might you suggest to Lena as an alternate explanation for her observations? 35. A box contains 48 tickets for six door prizes. Six tickets are drawn, and not replaced, to declare the winners. If Serge purchases 8 tickets, determine the probability that he wins a) the first prize b) the first and second prizes c) all six prizes 36. Elvira has three caps: one red, one green, and one blue. She also has two green T-shirts, one blue T-shirt, and three black T-shirts. Elvira considers it a fashion faux pas to ever be seen wearing blue and green at the same time. What is the probability that Elvira will commit a fashion faux pas, if she randomly reaches into her dresser and selects a cap and T-shirt? MDM4U Exam Review Package Name _______________________________ MDM4U Exam Dec 20 and 21, 2017 37. Ten boys and twelve girls decide to rent a 16-passenger van and a 6-passenger car to drive to a rock concert in a nearby city. If the group is distributed randomly between the vehicles, what is the probability that a) there are no boys in the car? b) there are no girls in the car? c) Steve and Susan are both in the van? d) either Steve or Susan are in the van? 38. Six friends at a dinner party are seated randomly at a round table. What is the probability that a) Russel will be seated next to Sadia? b) Russel, Jade, and Sadia are seated next to each other in alphabetical order, clockwise? 39. Suppose that whenever Enrico and Cliff play chess, odds are 2:1 in favour of Enrico winning a game. Suppose a 3game match is arranged. a) Create a tree diagram that illustrates all the possible outcomes for the match. b) Determine the probability of each outcome. c) Add the probabilities determined in part b). Does this total make sense? Explain. d) What is the probability that Enrico will not win all three games? e) What is the probability that Cliff will win exactly two games? 40. Determine if a uniform, binomial, or hypergeometric distribution would be the best model for each of the following experiments. Explain your reasoning. a) counting the number of spades in a hand of five cards dealt from a well-shuffled deck b) rolling a 2 on a die c) predicting the number of tails when flipping a coin 30 times 41. A lottery ticket costs $1.00 and a total of 2 250 000 tickets were sold. The prizes and their frequencies are given in the following table. Prize Number of Prizes $250 000 1 Determine the expected winnings of each ticket. $25 000 2 $2 500 5 $250 20 $25 100 42. Of 20 people invited to a pool party, 4 prefer vanilla ice cream, 7 prefer chocolate, and 3 prefer strawberry. The host surveys six of these people at random to determine how much ice cream to buy. a) What is the probability that at least 3 of the people surveyed prefer chocolate ice cream? b) What is the probability that none prefer vanilla ice cream? c) What is the expected number of people who prefer strawberry ice cream? d) What is the expected number of people who do not have a preference for any of the three flavours? 43. Suppose you randomly choose an integer, n, between 1 and 10, and then draw a circle with a radius of n centimetres. What is the expected area of this circle to the nearest hundredth of a square centimetre? 44. The probability of winning a prize in a game is 0.2. a) Show the probability distribution for the number of prizes won in 10 games. b) If the game will be played 1000 times during the fair, how many prizes should the game operators keep in stock? MDM4U Exam Review Package Name _______________________________ MDM4U Exam Dec 20 and 21, 2017 45. A multiple-choice mathematics quiz has five questions, each with five possible answers. Someone simply guesses at each answer. a) What is the probability of zero or one correct guesses? b) What is the probability of getting more than half the questions right? c) What is the expected number of correct guesses? 46. In a game of chance, a pair of dice is rolled. If the sum is 2 or 12, you win $10. If the sum is 6, 7, or 8 you win $1. You win nothing for any other sum. a) Create a probability distribution for this situation. b) Calculate the expected winnings for this game. c) How much should the dealer charge to play the game so that they make an average profit of 50 cents per game? 47. In the spring, the Ministry of the Environment caught and tagged 300 raccoons in an area in cottage country. The raccoons were released after being vaccinated against rabies. To estimate the raccoon population in the area, the ministry caught 25 raccoons during the summer. Of these, 10 had tags. Estimate the raccoon population in the area. 48. A cereal manufacturer finds that the mass of cereal in the 200-g packages is normally distributed with a mean of 200 g and a standard deviation of 16.3 g. a) What is the probability that a package chosen at random has a mass between 180 g and 220 g? b) If a container with less than 170 g is considered below standard, what proportion of cereal packages would be rejected? c) Out of 500 packages, how many have a mass greater than 230 g? 49. The probability of winning a large stuffed animal in the ring-toss game at a fair is 10%. a) Using a binomial distribution and a normal distribution, predict how the probabilities of winning at least 5 of 50 games will differ. b) Find the probability of winning at least 5 of 50 games using i) a binomial distribution ii) a normal approximation c) Do your calculations support your predictions? 50. How many bridge hands (13 cards) contain five clubs, two hearts, three diamonds, and three spades? Leave answer in factorial form. 51. A is the event of rolling a prime number with a die. B is the event of rolling a perfect square with a die. C is the event of rolling an even number with a die. Find :a) P(AB) b) P(CB) 52. In a manufacturing process, it is estimated that only 2% of the bolts that are machined are declared defective, that is, they are either too large or too small. In a package of 50 bolts, what is the probability that there is at least one defective bolt? 53. If test marks are assumed to be normally distributed and 70% of the students scored greater than a mark of 60 (out of 100), and 10% scored over 90, determine the mean and standard deviation of the test marks. 54. Explain each of the following types of cause and effect. Illustrate with an example. a) common-cause relationship b) accidental relationship c) cause and effect relationship d) reverse cause and effect e) presumed relationship MDM4U Exam Review Package Name _______________________________ MDM4U Exam Dec 20 and 21, 2017 55. Claire, a high school student, claims that listening to loud music helps her study. To defend her argument, she compiles the following results for four recent tests and her study habits: Test History English Mathematics Science 0 1 2 1.5 Volume of Music While Studying (Dial Setting) 48 59 72 70 Score (percent) a) Create a scatter plot for these data and classify the linear correlation. b) Do these data support Claire’s claim? Explain. c) To what extent has Claire established a cause and effect relationship? d) Identify at least two extraneous variables. e) Identify at least two types of bias that could be present in this study. Do you think that this bias could be intentional or unintentional? Explain. f) Suggest ways that Claire might improve the validity of her study. g) Determine the correlation coefficient and least-squares line of best fit for this data. 56. Students marks on a test were normally distributed with a mean of 70 and a standard deviation of 5. a) What percent of the students obtained a mark above 86%? b) What percent of the students obtained a B grade (70 to 80)? c) Determine the mark under which 75% of these students’ marks occur. (This is referred to as the 75th percentile.)
